Question

In the canonical quantization approach for QFT, we deal with operators & their (anti)commutation relations. However, at the same time, we say that the field operators are the solutions of equation of motions such as Dirac, Klein-Gordon etc.

In the path integral approach we don't deal with commutation relations of field operators by definition. So where do the equation of motions (of which field operators are solutions) go?

How can we have a physical theory without equations of motion? And as an example, what would the analogue of the Klein-Gordon equation be in the path integral approach?

Answer 1

In the path integral approach to quantum field theory (QFT), the focus is on probability amplitudes of different paths of the fields, rather than the commutation relations of field operators. The equation of motion, such as the Klein-Gordon equation, is still relevant in the path integral approach, but it is expressed in terms of the fields themselves rather than the operators. The analogue of the Klein-Gordon equation in the path integral approach can be derived from the action principle.

Step-by-step explanation:

In the canonical quantization approach for quantum field theory (QFT), operators and their (anti)commutation relations are used.

However, in the path integral approach, the focus is not on the commutation relations of field operators. Instead, the path integral approach deals with the probability amplitudes of different paths of the fields.

The equation of motion, such as the Dirac or Klein-Gordon equation, is still relevant in the path integral approach, but it is expressed in terms of the fields themselves rather than the operators.

The analogue of the Klein-Gordon equation in the path integral approach can be derived from the action principle.

It involves integrating over all possible field configurations and assigning a weight to each configuration based on the action functional.